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Functions, operator => eigenfunction, eigenvalue

by sundriedtomato
Tags: eigenfunction, eigenvalue, functions, operator
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sundriedtomato
#1
Sep24-07, 05:57 AM
P: 12
1. The problem statement, all variables and given/known data
Show, that functions
f1 = A*sin([tex]\theta[/tex])exp[i[tex]\phi[/tex]] and
f2 = B(3cos[tex]^{2}[/tex]([tex]\theta[/tex]) - 1) A,B - constants
are eigenfunctions of an operator

and find eigenvalues


3. The attempt at a solution
This is what i got for the first function:


The next step is to solve left part of the equation, and than compare it to the right part.

The question arises, how to solve that equation?

I tried simplifying left part of an equation in mathcad, and I got


Next question from that part, is if I am doing it right, how to compare those parts, and answer a question - weather this function is an eigenfunction of an operator?

Thank You in advance, and I am constantly near computer and waiting for suggestions.
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Reshma
#2
Sep24-07, 09:24 AM
P: 777
Your eigen operator has partial derivatives wrt to [itex]\theta[/itex] and [itex]\phi[/itex]. When you operate it on your given eigen function, you should get back your original function multiplied by a scaling factor which is your eigenvalue.
sundriedtomato
#3
Sep24-07, 11:25 AM
P: 12
Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?

nrqed
#4
Sep24-07, 11:32 AM
Sci Advisor
HW Helper
P: 2,887
Functions, operator => eigenfunction, eigenvalue

Quote Quote by sundriedtomato View Post
Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?
Just go ahead and apply the derivatives!! It's that simple. (btw, I don't know what you entered in mathcad but what it gave you is wrong).

all you have to do is to apply the derivatives
nrqed
#5
Sep24-07, 11:34 AM
Sci Advisor
HW Helper
P: 2,887
Quote Quote by sundriedtomato View Post
Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?
For the first term, what you have to calculate is

[tex] \frac{1}{sin \theta} \frac{\partial}{\partial \theta} ( sin \theta ~\frac{\partial}{ \partial \theta} (A sin \theta} e^{i \phi}}) ) [/tex]
sundriedtomato
#6
Sep24-07, 12:37 PM
P: 12
I will give it a try right know. Thank You.
sundriedtomato
#7
Sep24-07, 01:58 PM
P: 12
So, after proper calculations, the result is


Is this one correct?

I posted an image of what I am given, and as far as I know, differentiation sign usually is placed before the function?
I just don't get it - to what parts of an equation do underlined derivatives belong t?


Thank You.
rahuldandekar
#8
Sep24-07, 09:27 PM
P: 69
The part in brackets is an "operator"... every incomplete differentiation sign (ie, without anything to differentiate) operates on whatever is "multiplied" to it.

For example, the d/d(theta) is your image also operates on A*sin(theta)*exp(i*phi), because when you open the bracket, it gets to differentiate that term.

Eg. (the d's are partial)
[d/dx + d/dy]*x*y^2 = y^2 + 2*y*x
Reshma
#9
Sep25-07, 08:21 AM
P: 777
Quote Quote by sundriedtomato View Post
Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?
Just operate it on the function. They are simple partial derivatives, I won't take much time to solve. Just a little patience for the initial steps. I tried your problem and many terms get canceled out and you get the solution correctly. You don't need mathcad to do it.
sundriedtomato
#10
Sep25-07, 11:04 AM
P: 12
Thank You Reshma! I already managed to solve it correctly, and find eigenvalues as well.
For the case with the first function : a = 2h^2, and for the case with second function b = 6h^2. The problem was initially in understanding how to apply operator properly. Once example have been given, and operator properties reanalyzed - problem got solved in like 10 minutes (with putting it all on a paper as well). Thank You everybody who took part in this one!


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